(a) For a dilute gas, the pair correlation function \(g(r)\) may be approximated as

\[g(r) \simeq \exp \{-u(r) / k T\}\]

Show that, under this approximation, the virial equation of state (10.7.11) takes the form

\[\frac{P V}{N k T} \simeq 1-2 \pi n \int_{0}^{\infty} f(r) r^{2} d r\]

where \(f(r)[=\exp \{-u(r) / k T\}-1]\) is the Mayer function, equation (10.1.6).

(b) What form will this result take for a gas of hard spheres? Compare your result with that of Problem 1.4.